DELHI PUBLIC SCHOOL NEWTOWN

SESSION 2023–24
FINAL EXAMINATION

CLASS: IX FULL MARKS: 80

1
SUBJECT: MATHEMATICS [SET A] TIME: 22 HOURS

Answers to this Paper must be written on a paper provided separately.

You will not be allowed to write during the first 15minutes.
This time is to be spent in reading the question paper.
The time given at the head of this Paper is the time allowed for writing the answers.
Attempt all questions from Section A and any four questions from Section B.
The intended marks for questions or parts of questions are given in brackets [ ].
This paper consists of 5 printed pages.

SECTION A
(Attempt all questions from this section)

Question1

Choose the correct answers to the questions from the given options. (Do not copy the question,
Write the correct answer only.) [15]

(i) The greatest among 2√𝟔, 6√𝟐, √𝟏𝟐 and 4√𝟑 is:

(a) 2√𝟔 (b) 6√𝟐 (c) √𝟏𝟐 (d) 𝟒√𝟑

(ii) The compound interest on ₹ 1200 for two years at the rate of 5% p.a. compounded
annually is:
(a) ₹ 60 (b) ₹ 120 (c) ₹ 123 (d) ₹ 125

(iii) Factors of x2 – (y - z ) 2 are :

(a) (x2 + y2) (b) (x + y + z)(x -y – z) (c) (x + y - z)(x - y + z) (d) 4xy

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 𝟏
(iv) The square of 𝒂 + is:
𝒂

𝟏 𝟏 𝟏 𝟏
(a) 𝒂𝟐 + 𝒂𝟐 + 𝟏 (b) 𝒂𝟐 + 𝒂𝟐 + 𝟐 c) 𝒂𝟐 + 𝒂𝟐 d) 𝒂𝟐 − 𝒂𝟐

(v) The value of log 30 - log 3 is:

a) 0 b) 1 c) 2 d) -1

(vi) The perimeter of a square, if its diagonal is 10 √ 2 cm is:

a) 10 cm b) 40√2 cm c) 20 cm d) 40 cm

(vii) If a ladder 13m long reaches a window 5m above the ground, then the distance of the foot
of the ladder from the base of the wall is:

a) 18m b) 12m c) 16m d) 14m

(viii) The line segments joining the midpoints of consecutive sides of a parallelogram form a:

a) triangle b) kite c) trapezium d) parallelogram

(ix) A path of 7 m wide runs around outside a circular park whose radius is 18 m, the diameter
of the outer circle is:

a) 53m b) 70m c) 25m d) 50m

𝑨+𝑩
(x) In a ∆ ABC Sin ( ) will be:
𝟐

𝑪 𝑪 𝑨+𝑩 ( 𝑨 + 𝑩)
a) cos 𝟐 b) sin 𝟐 c) sin 2 ( ) d) cos
𝟐 𝟐

(xi) The coordinate of the point whose ordinate is - 4 and lies on y-axis is:

a) (0 ,-4) b) (0, 4) c) ( -4, 0) d) (4, -4)

(xii) The triangle and a rhombus are on the same base and between the same parallels, the ratio
of the area of the triangle to the area of the rhombus is:

a) 1: 3 b) 3 : 2 c) 4 : 1 d) 1: 2

(xiii) The lateral surface area of a cube is 256 m², its volume is:

a) 512 m3 b) 216 m3 c) 64 m3 d) 256 m3

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(xiv) The following marks obtained by a student in a test are 81, 72, 90, 90, 85, 72, 86, 92, 70, 81,
83, 92, 85, 89 ,79, 62. The range is:

a) 33 b) 25 c) 30 d) 35

(xv) Study the figure and answer the following:

Assertion (A): From the given figure if x < y then the correct relation is BC > AC.
Reason (R): If two sides of a triangle are of length 4 cm and 2.5 cm then the third side cannot
be 5.5 cm.
a) A is true, R is false b) A is false, R is true
c) Both A and R are true d) both A and R are false

Question 2

(i) Calculate the compound interest accrued on ₹ 6,000 in 3 years, compounded yearly, if the
rates for the successive years are 5%, 8% and 10% respectively. What will be the total amount
after 3 years? [4]

(ii) Find the perimeter of a rectangle whose diagonal is 29 cm and one of its sides is 20 cm. [4]

(iii) Construct a parallelogram ABCD with side AB = 6.5 cm, BC = 4.5 cm and ∠ABC = 60° [4]

Question 3
(i) In the figure, PQRS is a square and SRT is an equilateral triangle. Prove that PT = QT.
[4]

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(ii) From the given figure find a) cos y° b) tan x° [4]

(iii) Construct a frequency polygon for the following data. [use graph paper] [5]
Age in years 10-20 20-30 30-40 40-50 50-60 60-70
No. of Persons 3 6 8 9 5 2

SECTION B
(Attempt any four questions from this section)

Question 4
𝟐 𝟐 𝟐
(i) If ( )−𝟔 × ( ) 𝟑 = ( ) 𝟐𝒙−𝟏 , find the value of x. [3]
𝟕 𝟕 𝟕

(ii) The altitude AD of ∆ ABC is 12 cm long. If BD is 8 cm and DC is 18 cm, prove that ∠BAC is
right angle. [3]

(iii) Abhay bought stamps of denominations ₹ 2 and ₹ 5 for ₹ 76. If he bought 20 stamps in all,
find the number of stamps of each type. [4]

Question 5
𝟏
(i) If tan A = and tan B = √𝟑, find the value of sin A cos B + Cos A sin B [3]
√𝟑

(ii) On what sum of money does the difference between the simple interest and compound interest
for two years at 5% amount to ₹ 15? [3]

(iii) A cubical tank of side 21 m contains water. A cube of side 10.5 m is completely merged into
the water. Calculate the rise in the water level. (corrected up to two decimal places) [4]

Question 6
(i) Solve for x when 𝐥𝐨𝐠 𝟑 ( 𝐥𝐨𝐠 𝟐 𝟐𝒙 )= 𝐥𝐨𝐠 𝟓 + 𝐥𝐨𝐠 𝟐 [3]

(ii) In triangle ABC, D is a point on BC (in between B and C). Prove that
AB + BC + AC > 2AD. [3]

(iii) In a trapezium ABCD, sides AB and DC are parallel to each other and E is the midpoint of
AD. A line through E is drawn parallel to AB which intersects BC at F. Prove that
AB + DC = 2EF. [4]

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Question 7
(i) If the volume of a room is 792 m³ and the area of the floor is 132 m², find the lateral surface
area of the room when sum of its length and breadth is 28 m. [3]

(ii) In figure, PQRS and ABRS are parallelograms and X is any point on side BR. Show that
(a) ar (PQRS) = ar (ABRS)
𝟏
(b) ar (AXS) = 𝟐 ar (PQRS) [3]

(iii) The radius of a circle with centre at O is 13 cm. AB and CD are two parallel chords. The
length of AB is 24 cm and distance between the chords is 17 cm, find the length of CD. [4]

Question 8
(i) Factorise: 11x2 + 33x – 110 [3]

(ii) Prove that the points with coordinates (7, 10), (-2, 5) and (3, -4) are the vertices of an isosceles
triangle. [3]

𝟏
(iii) If x = √3 - √2, find the value of x2 − [4]
𝒙𝟐

Question 9
𝟏
(i) If 2(x2 + 1) = 5x, find the value of x2 + . [3]
𝒙𝟐

(ii) Prove that any point on the external angle bisector of a triangle is equidistant from the sides
containing the angle. [3]

(iii) A race track is in the form of a ring, whose inner and outer circumferences are 88 m and
132 m respectively. Find the width of the track and also, it’s area. [4]

Question 10
(i) The sides of a triangular field are 15 m, 14 m and 13 m, Find the area of the triangle. [3]

(ii) The following observations 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95 have been arranged in
ascending order. If the median of the data is 63, find the value of x. [3]

(iii) ABCD is a square and P is point inside it. ∆ABP is an equilateral triangle. Find the
measure of ∠ APB, ∠ BPC, ∠ PCD and reflex ∠APC. [4]

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